It's actually quite a popular way of doing things, I personally find it useful particularly with Crib, Lap and Baby sized quilts, I've not yet done it on anything larger.
I actually didn't ever do it this way until I started doing charity quilting a few years back. A pattern had been provided to me without sufficient fabric for binding (but had a larger backing) and the pattern itself suggested that I use the birthing method and described how to do it. Sometimes, it can be a very time saving and effective design that way. Specifically with charity quilting, you can get more done this way, so it is a method which really allows me to give more back when I have the time to make them.
One thing though in my experiences, (hopefully this makes sense to you) - I always make two to three rounds of sewing "around" - dependant upon the fabric, piecing etc. Once around for the batting and Top and another for the backing to be attached then once more prior to birthing because when only doing the one simple zip around of all the layers, seam allowances have tended to fray/tear away from the action of pulling the quilt right side out - even when I tried to be very gentle. The reenforcing I now do when using the birthing method was a tip I learned watching Sue Hausmann who'd had similar problems and was discussing solutions for it. I've never had that problem again now that I do my reinforcements as she had suggested and the quilts come out just great!
Explosive blessings, abundance and inspiration to you all!
- The Creative Seamstress